Engineering Graphics, Class 10Points, Lines and Planes

Mechanical Engineering Department

Introduction

Points, lines and planes are the basic geometric elements used in three-dimensional spatial geometry, called descriptive geometry.

An understanding of how to locate and manipulate these elements in their simplest form is basic to the graphical analysis of problem encountered in a number of engineering applications, including intersections, developments and others.

Points in space

A point is a theoretical location in space having no dimension. A point in space may be defined by its coordinates from a fixed reference. P = (x,y,z).

In descriptive geometry, we use plane sheets of paper to solve three dimensional problems graphically. To be able to represent spatial point on the plane of the paper, we use the glass box concept.

Projection of a point

Given a point P = (x,y,z), projecting P into the planes of the glass box shows its x and z coordinates in the front plane, its x and y coordinates in the top plane and its y and z coordinates in the profile plane. Unfolding the glass box provides a complete description of the point’s location in the 3D space.

The Folding Line

The folding line is the intersection between two projection planes

The folding line TF between the top and the front views is the intersection of the horizontal and frontal planes. Folding line FR, between the front and the right-side views is the intersection of the frontal & the right side planes.

Point projection constraints

Since the x coordinate of a point appears in both the frontal view and the top view, its projections in those two views must be horizontally aligned.

Since the z coordinate of a point appears in both the frontal view and the right side view, its projection in those two views must be vertically aligned.

Since the y coordinates of a point appears in both the top view and the right side view, the distance of the projections from the folding lines must be the same.

Projection of a Point

Note that any two of the glass box projections are sufficient to completely describe the three independent coordinates of a point in space.

In descriptive geometry, only the projections of the point are usually shown. The coordinate lines are shown here for illustration purposes.

Given the projection of a point in two of the planes, its projection in the third plane can easily be determined.

Finding the missing projection of a point

Given the top and the right side projections of a point, the front projection is found by erecting perpendiculars to the TF and FR folding lines. The frontal projection is at the intersection of the two perpendiculars.

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Finding the missing projection of a point

Given the frontal and the top projections of a point, the right side projection is found by dropping a perpendicular to the FR folding line from the frontal projection, and measuring a distance from the FR folding line equal to the distance of the top projection from the TF folding line.

Finding the missing projection of a point

Given the frontal and the right side projections of a point, the top projection is found by dropping a perpendicular to the TF folding line from the frontal projection, and measuring a distance from the TF folding line equal to the distance of the right side projection from the FR folding line.

Moving the folding lines

Moving the profile projection plane will result in moving the FP folding line. The equality of the distance y in the top and right side views is not affected.

You may place the folding lines at any convenient location, provided the equality relationship between corresponding distances is respected.

The folding line in auxiliary views

In auxiliary views, the positions of the folding lines depend upon the positions of the planes of the glass box with respect to the point.

The distance y is the same in all of the auxiliary views

The folding line in auxiliary views

Given two views, the projection of a point in any auxiliary view can be easily determined by using the two following rules:

In any two consecutive views, the projections are aligned with the perpendicular to the folding lines.

In any three consecutive views, the distance of the projection from the folding line in the first view is equal to the distance of the folding line in the third view

Locating a point in an auxiliary view (skip-a-view rule)

Based on the two previous rules, the location of a point in an auxiliary view can be determined given its projection in two views

Extend projection lines into the auxiliary views that are perpendicular to the folding lines.

Transfer the distance of the point from the fold line in the preceding view to locate its projection in the new view.

The preceding method is called the skip-a-view rule. This rule has very useful applications in descriptive geometry

Line projection

A line is the straight path between two points

The projection of a line into the front, top and right side planes is defined by the projection of its endpoints onto these planes.

When a line is projected into a projection planes it appears foreshortened (shorter length), unless it is parallel to the projection plane. In this case, it appears in its true length.

Point’s visibility (front and back)

The front view by itself does not tell which point is in the front. This must be deduced from the top and right side projections.

The point in the front appears closer to the TF folding line in the top projections, and closer to the FR folding line in the right side projection. In the example shown, point b is in front of point a.

Point’s visibility (up and down)

The top view by itself does not tell which point is in the top. This must be deduced from the front and right side projections.

The point in the top appears closer to the TF folding line in the front projections, and closer to the TR folding line in the right side projection. In the example shown, point b is on top of point a.

Point’s visibility (right and left)

The right side view by itself does not tell which point is to the right. This must be deduced from the front and top projections.

The point to the right appears closer to the FR folding line in the front projections, and closer to the TR folding line in the top projection. In the example shown, point b is to the right of point a.

Point’s visibility (General Rule)

In general, the closer a point’s projection is to the folding line in certain view, the more “priority in visibility” it has in the adjacent view.

Horizontal principal lines

Principal lines are parallel to at least one of the principal projection planes. A horizontal principal line is parallel to the horizontal (top) projection plane.

A horizontal principal line appears in true length in the horizontal (top) view.

A horizontal principal line appears parallel to the TF folding line in the front view and parallel to the TR folding line in the right side view.

Frontal principal lines

A frontal principal line is parallel to the frontal projection plane, and appears in true length in the frontal view.

A frontal principal line appears parallel to the TF folding line in the top view and parallel to the FR folding line in the right side view.

Profile principal lines

A profile principal line is parallel to the profile (right side) projection plane, and appears in true length in the right side view.

A frontal principal line appears parallel to the FR folding line in the front view and parallel to the TR folding line in the top view.

Principal lines (General rules)

A principal lines is parallel to at least one of the principal projection planes

A principal line appears in true length in the principal projection plane to which it is parallel, and appears parallel to the folding line in the adjacent views.

A line will appear in its true length in a view taken such that that the fold line is parallel to the current projection (True length rule).

Perpendicular lines

A line that is perpendicular to one of the principal projection planes will appear as a point in that plane. This projection is called a point view or an end view of the line.

A perpendicular line will be a principal lines for all the projection planes normal to the plane on which it is perpendicular, and will appear in true length in those projection planes.

Rule 1: The alignment rule

The projections of a point in two consecutive views are aligned with respect to the normal to the folding line.

Rule 2: Skip-a-view

The distance between the projection of a point and the folding line is equal in any two views in three consecutive views with one view skipped

Rule 3: True length of line

A line will appear in its true length in a view taken such that that the fold line is parallel to the current projection.

Rule 4: Point view (end view) of line

If a line appears as a point view in a plane, it will appear in its true length in all planes normal to that plane.

Rule 5: Visibility

The closer a point’s projection is to the folding line in certain view, the more “priority in visibility” it has in the adjacent view.